Aspects of Symmetry, Topology and Anomalies in Quantum Matter

TL;DR

This thesis develops a unified framework linking symmetry, topology, and anomalies in quantum matter across condensed-matter and high-energy perspectives. It introduces symmetry-twist and wavefunction-overlap strategies, a spacetime-path-integral cocycle program, and lattice constructions to classify and realize SPTs and topological orders, including nonperturbative approaches to chiral fermions on lattices. A central theme is the bulk-boundary correspondence: boundary anomalies reflect bulk topology, and boundary gapping rules correspond to ’t Hooft anomaly matching, even in 3+1D twisted gauge theories with SL(3,Z) data. The work provides concrete classifications via group cohomology, explicit lattice models for anomaly-free chiral matter, and a leading-edge treatment of higher-dimensional topological orders, with implications for nonperturbative formulations of chiral gauge theories. Collectively, these results offer a rigorous, multi-faceted toolkit for understanding emergent quantum phases and their topological invariants, with potential applications to quantum computation and beyond.

Abstract

In this thesis, we explore the aspects of symmetry, topology and anomalies in quantum matter with entanglement from both condensed matter and high energy theory viewpoints. The focus of our research is on the gapped many-body quantum systems including symmetry-protected topological states and topologically ordered states. Chapter 1. Introduction. Chapter 2. Geometric phase, wavefunction overlap, spacetime path integral and topological invariants. Chapter 3. Aspects of Symmetry. Chapter 4. Aspects of Topology. Chapter 5. Aspects of Anomalies. Chapter 6. Quantum Statistics and Spacetime Surgery. Chapter 7. Conclusion: Finale and A New View of Emergence-Reductionism. (Thesis supervisor: Prof. Xiao-Gang Wen)

Figures (42)

• Figure 1: Quantum matter: The energy spectra of gapless states, topological orders and symmetry-protected topological states (SPTs).
• Figure 2: (a) Gilzparg-Wilson fermions can be viewed as putting gapless states on the edge of a nontrivial SPT state (e.g. topological insulator). Our approach can be viewed as putting gapless states on the edge of a trivial SPT state (trivial insulator) and introduce proper strong interactions to gapped out the mirror sector (in the shaded region). (b) The equivalence of the boundary gapping criteria and the 't Hooft anomaly matching conditions. Our proof is based on a bulk theory of Abelian SPT described by a $K$-matrix Chern-Simons action $\frac{K_{IJ}}{4\pi} \int a_{I} \wedge \space\mathrm{d} a_{J}$. A set of anyons, labeled by a matrix $\mathbf{L}$, with trivial mutual and self statistics is formulated as $\mathbf{L}^T \cdot K^{-1} \cdot \mathbf{L}=0$. This condition is equivalent to a 1-loop anomaly-matching condition for fermions, or more generally as ${{\mathbf{t}^T} K {\mathbf{t}}=0}$ for both bosons and fermions, where $\mathbf{t}$ is a matrix formed by the charger coupling between matter fields and external gauge fields (as solid lines and wavy lines respectively in the Feynman diagram).
• Figure 3: (a) The square lattice toric code model with $A_v$ and $B_p$ operators. (b) The $e$-string operator with end point $e$-charge ($Z_2$ charge) excitations on the vertices, created by a product of $\prod \sigma_z$. The $m$-string operator with end point $m$-charge ($Z_2$ flux) excitations in the plaquette, created by a product of $\prod \sigma_x$. See an introduction to toric code in Kitaev:1997wrpachos2012introduction.
• Figure 4: The illustration for ${\mathsf{O}}_{\text{(A)(B)}}= \langle \Psi_{\text{A}} | \hat{\mathsf{O}} | \Psi_{\text{B}}\rangle$. Evolution from an initial state configuration $|\Psi_{in} \rangle$ on the spatial manifold (from the top) along the time direction (the dashed line - - -) to the final state $|\Psi_{out} \rangle$ (at the bottom). For the spatial $\mathbb{T}^{d}$ torus, the mapping class group MCG$(\mathbb{T}^{d})$ is the modular SL$(d,\mathbb{Z})$ transformation. We show schematically the time evolution on the spatial $\mathbb{T}^{2}$, and $\mathbb{T}^{3}$. The $\mathbb{T}^{3}$ is shown as a $\mathbb{T}^{2}$ attached an $S^1$ circle at each point.
• Figure 5: $\hat{S}$-move is $90^\circ$ rotation. We apply the symmetry-twist along $x$ and $y$ axis, where $h_x$ and $h_y$ are the twisted boundary condition assigned respect to its codimension directions. (a) A system on $T^2$ with $h_x$ and $h_y$ symmetry twists. Here $T^2$ has the same size in $x$ and $y$ directions in order to have meaningful wavefunction overlap. (b) The resulting symmetry twists after the $\hat{S}$-move.